Deriving test cases from a truth table without testing every row

A truth table is a complete list of test inputs with expected results; the question is which rows to keep. For the loan rule below, four conditions and sixteen rows, the engine's independence pairs pick five rows that show every condition's effect, and the Python export turns any row into an executable check.

The rule is income_ok AND (credit_ok OR guarantor) AND NOT bankrupt. The full table has sixteen rows, each a valid test with an expected result computed from the rule. Testing all sixteen is reasonable here and hopeless at eight conditions, so the method is to keep the rows that carry information. The independence pairs are: income_ok, from no income with a guarantor (ineligible) to income with a guarantor (eligible); credit_ok, from income with neither credit nor guarantor (ineligible) to credit (eligible); guarantor, from the same base to a guarantor (eligible); bankrupt, from eligible with a guarantor to bankrupt (ineligible).

Five distinct rows cover the four pairs. Each row is a complete assignment and an expected outcome, so each is a test case; the CSV export writes them with the rest of the table, and the Python export gives a function that evaluates the rule and rejects any input that is not exactly True or False, which is what a parameterised test can call.

Rows to add on top of the pairs are the ones that encode a requirement rather than a condition: the all-FALSE row, the all-TRUE row (bankrupt wins), and any counterexample a previous version of the rule produced. The regression page and the counterexamples-as-tests page cover those two sources.

Example

eligible := income_ok AND (credit_ok OR guarantor) AND NOT bankrupt · output

Exact checks

Always true (tautology)? No Counterexample: income_ok = F, credit_ok = F, guarantor = F, bankrupt = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: income_ok = T, credit_ok = F, guarantor = T, bankrupt = F · Exhaustively checked and solver cross-checked · 11 rows

Can be true? Yes Example assignment: income_ok = T, credit_ok = F, guarantor = T, bankrupt = F · Exhaustively checked and solver cross-checked · 11 rows

Truth table (16 rows)

Deriving test cases from a truth table without testing every row
#	income_ok	credit_ok	guarantor	bankrupt	eligible
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	FALSE	TRUE	FALSE
2	FALSE	FALSE	TRUE	FALSE	FALSE
3	FALSE	FALSE	TRUE	TRUE	FALSE
4	FALSE	TRUE	FALSE	FALSE	FALSE
5	FALSE	TRUE	FALSE	TRUE	FALSE
6	FALSE	TRUE	TRUE	FALSE	FALSE
7	FALSE	TRUE	TRUE	TRUE	FALSE
8	TRUE	FALSE	FALSE	FALSE	FALSE
9	TRUE	FALSE	FALSE	TRUE	FALSE
10	TRUE	FALSE	TRUE	FALSE	TRUE
11	TRUE	FALSE	TRUE	TRUE	FALSE
12	TRUE	TRUE	FALSE	FALSE	TRUE
13	TRUE	TRUE	FALSE	TRUE	FALSE
14	TRUE	TRUE	TRUE	FALSE	TRUE
15	TRUE	TRUE	TRUE	TRUE	FALSE

Plain-language reading

Beginnereligible is true exactly when all of the following hold:
1. income_ok is true
2. at least one of credit_ok or guarantor is true (both may be true)
3. bankrupt is false
In every other case, eligible is false.
About the operators:
- AND is true only when all of its parts are true.
- OR is inclusive: it is true when at least one part is true, including when several are.
- NOT reverses a value: true becomes false and false becomes true.

Intermediateeligible is true exactly when all of the following hold:
1. income_ok is true
2. at least one of credit_ok or guarantor is true (both may be true)
3. bankrupt is false
In every other case, eligible is false.

Advancedeligible is true exactly when all of the following hold [conjunction ∧]:
1. income_ok is true
2. at least one of credit_ok or guarantor is true (both may be true) [inclusive disjunction ∨]
3. bankrupt is false [negation ¬]
In every other case, eligible is false.
Formal notation: income_ok ∧ (credit_ok ∨ guarantor) ∧ ¬bankrupt

Logic diagram

MC/DC

Input	Status	Independence pair (only that input changes)
income_ok	found	income_ok=F, credit_ok=F, guarantor=T, bankrupt=F → F \| income_ok=T, credit_ok=F, guarantor=T, bankrupt=F → T
credit_ok	found	income_ok=T, credit_ok=F, guarantor=F, bankrupt=F → F \| income_ok=T, credit_ok=T, guarantor=F, bankrupt=F → T
guarantor	found	income_ok=T, credit_ok=F, guarantor=F, bankrupt=F → F \| income_ok=T, credit_ok=F, guarantor=T, bankrupt=F → T
bankrupt	found	income_ok=T, credit_ok=F, guarantor=T, bankrupt=F → T \| income_ok=T, credit_ok=F, guarantor=T, bankrupt=T → F

The same rule as code

Python

# -*- coding: utf-8 -*-
# Generated by Tangibool (tangibool.python/1). Do not edit by hand; regenerate from the source revision.
# Model: not stated
# Source revision: reference
# Interpretation status: not stated
"""Tangibool evaluator (decision model).

Generated deterministically from the validated Boolean model; it evaluates exactly that
model. It does not establish that the model is a complete, legal, or safe statement of a
real-world rule, and evaluating examples does not verify the rule.

Source revision: reference
Interpretation status: not stated
Exporter: 1.0.0 (format tangibool.python/1; engine 1.0.0; semantics tangibool-semantics/1)
Logic digest: 40aeadff132d113e9694381e8ed2530396d9cd296f720e1c4c08565e8f3b69f4
Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78

Inputs (dict keys, declared order) and generated parameter names:
    'income_ok' -> input_income_ok
    'credit_ok' -> input_credit_ok
    'guarantor' -> input_guarantor
    'bankrupt' -> input_bankrupt

Outputs (declared order) and their functions:
    'eligible' -> output_eligible(inputs)

Assumptions (the declared scope, checked by in_scope(inputs)):
    none; every complete assignment is in scope.

Input contract:
    Every required input must be exactly True or False (type(value) is bool). Missing
    inputs, None, strings, integers (including 0 and 1), and keys that are not declared
    inputs raise InputError before anything is evaluated, including inputs that a
    short-circuiting operator would never read. Results are always bool.

Scope:
    Decision functions raise OutOfScopeError when the inputs violate an assumption; an
    out-of-scope assignment is never turned into a False result. Pass check_scope=False
    to evaluate without the scope check, and use in_scope(inputs) to test it separately.

Usage:

    import tangibool_model as model

    model.evaluate({'income_ok': False, 'credit_ok': False, 'guarantor': False, 'bankrupt': False})
    # -> {'eligible': False}
    model.output_eligible({'income_ok': False, 'credit_ok': False, 'guarantor': False, 'bankrupt': False})
    # -> False
    model.evaluate({'income_ok': 1, 'credit_ok': False, 'guarantor': False, 'bankrupt': False})
    # raises InputError: 1 is an int, not a bool
"""

import collections.abc as _abc
from typing import Dict, Mapping, Tuple

__all__ = [
    "MODEL_INFO",
    "IDENTIFIER_MAP",
    "INPUT_NAMES",
    "OUTPUT_NAMES",
    "ASSUMPTIONS",
    "InputError",
    "OutOfScopeError",
    "minterm_index",
    "in_scope",
    "evaluate",
    "output_eligible",
]

MODEL_INFO = {
    "format": "tangibool.python/1",
    "mode": "decision",
    "model": None,
    "revision": "reference",
    "interpretation": None,
    "exporter": "1.0.0",
    "engine_version": "1.0.0",
    "semantics_version": "tangibool-semantics/1",
    "logic_digest": "40aeadff132d113e9694381e8ed2530396d9cd296f720e1c4c08565e8f3b69f4",
    "assumptions_digest": "d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78",
    "primary_output": "eligible",
    "unused_inputs": (),
    "subcircuits": (),
}

# Stable ID -> generated identifier (reversible mapping; names are exact, escaped strings).
IDENTIFIER_MAP = {
    "v_6cab253eb6ee": {"kind": "input", "name": "income_ok", "identifier": "input_income_ok"},
    "v_d61ccf3c5e24": {"kind": "input", "name": "credit_ok", "identifier": "input_credit_ok"},
    "v_a90721d191bc": {"kind": "input", "name": "guarantor", "identifier": "input_guarantor"},
    "v_c9e9282e85b4": {"kind": "input", "name": "bankrupt", "identifier": "input_bankrupt"},
    "f_a3631a5ff6db": {"kind": "output", "name": "eligible", "identifier": "output_eligible"},
}

INPUT_NAMES = ("income_ok", "credit_ok", "guarantor", "bankrupt")
OUTPUT_NAMES = ("eligible",)
ASSUMPTIONS = ()

_DECLARED_INPUTS = frozenset(INPUT_NAMES)
_SCOPE_INPUTS = ()
_ASSUMPTION_FORMULAS = {entry["name"]: entry["formula"] for entry in ASSUMPTIONS}


class InputError(ValueError):
    """Raised when inputs are missing, undeclared, or not exactly True/False.

    Every required input must satisfy type(value) is bool. Missing inputs, None, strings,
    integers (including 0 and 1), and keys that are not declared inputs are rejected before
    anything is evaluated. A missing input is never treated as False. The individual problems
    are listed in the problems attribute.
    """

    def __init__(self, problems):
        self.problems = tuple(problems)
        super().__init__("Invalid inputs: " + "; ".join(self.problems))


class OutOfScopeError(ValueError):
    """Raised when the inputs violate the model's assumptions (its declared scope).

    Outside its assumptions the rule has no decision: this error is never a False result.
    The names of the violated assumptions are in the violated attribute.
    """

    def __init__(self, violated):
        self.violated = tuple(violated)
        details = ", ".join(name + " (" + _ASSUMPTION_FORMULAS.get(name, "?") + ")" for name in self.violated)
        super().__init__(
            "Out of scope: the inputs violate "
            + ("assumption " if len(self.violated) == 1 else "assumptions ")
            + details
            + ". The rule has no decision outside its assumptions; this is not a False result."
        )


def _short_repr(value: object) -> str:
    text = repr(value)
    return text if len(text) <= 60 else text[:57] + "..."


def _checked_inputs(inputs: Mapping[str, bool], required: Tuple[str, ...], check_scope: bool) -> Dict[str, bool]:
    """Validate inputs strictly before anything is evaluated and return a plain copy."""
    if type(check_scope) is not bool:
        raise TypeError("check_scope must be True or False")
    if not isinstance(inputs, _abc.Mapping):
        raise InputError(["inputs must be a mapping from input names to True/False, got " + type(inputs).__name__])
    problems = []
    for key in inputs:
        if type(key) is not str or key not in _DECLARED_INPUTS:
            declared = ", ".join(repr(name) for name in INPUT_NAMES) or "none"
            problems.append("undeclared input " + _short_repr(key) + " (declared inputs: " + declared + ")")
    needed = tuple(required) + (_SCOPE_INPUTS if check_scope else ())
    for name in needed:
        if name not in inputs:
            problems.append("missing required input " + repr(name))
    for key, value in inputs.items():
        if type(key) is str and key in _DECLARED_INPUTS and type(value) is not bool:
            problems.append(
                "input " + repr(key) + " must be True or False (type bool), got "
                + type(value).__name__ + " " + _short_repr(value)
            )
    if problems:
        raise InputError(list(dict.fromkeys(problems)))
    return dict(inputs.items())


def _require_bool_args(function: str, pairs: Tuple[Tuple[str, object], ...]) -> None:
    problems = [
        "argument " + repr(name) + " of " + function + " must be True or False (type bool), got "
        + type(value).__name__ + " " + _short_repr(value)
        for name, value in pairs
        if type(value) is not bool
    ]
    if problems:
        raise InputError(problems)


def minterm_index(*values: bool) -> int:
    """Row index of an assignment over a declared variable order.

    The first value is the most significant bit, so all-False is row 0. Used for indexed
    minterm (sum) and maxterm (product) forms.
    """
    index = 0
    for value in values:
        if type(value) is not bool:
            raise InputError(["minterm_index() takes True/False values, got " + type(value).__name__ + " " + _short_repr(value)])
        index = index * 2 + (1 if value else 0)
    return index


def _scope_values(values: Dict[str, bool]) -> Tuple[Tuple[str, bool], ...]:
    """Value of every enabled, confirmed assumption for validated inputs, in declared order."""
    return ()


def _require_in_scope(values: Dict[str, bool]) -> None:
    violated = tuple(name for name, holds in _scope_values(values) if not holds)
    if violated:
        raise OutOfScopeError(violated)


def in_scope(inputs: Mapping[str, bool]) -> bool:
    """True when the inputs satisfy every assumption (the model's declared scope).

    Requires the inputs the assumptions read; raises InputError for invalid inputs.
    """
    values = _checked_inputs(inputs, (), True)
    return all(holds for _, holds in _scope_values(values))


# ---- Formula bodies (inputs are validated by the public functions below) ----


def _compute_output_eligible(input_income_ok: bool, input_credit_ok: bool, input_guarantor: bool, input_bankrupt: bool) -> bool:
    """Decision output 'eligible': income_ok AND (credit_ok OR guarantor) AND NOT bankrupt"""
    return input_income_ok and (input_credit_ok or input_guarantor) and (not input_bankrupt)


# ---- Public API ----


def output_eligible(inputs: Mapping[str, bool], *, check_scope: bool = True) -> bool:
    """Decision output 'eligible' (stable ID f_a3631a5ff6db).

    'eligible' := income_ok AND (credit_ok OR guarantor) AND NOT bankrupt

    Required inputs: 'income_ok', 'credit_ok', 'guarantor', 'bankrupt'.
    Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
    when check_scope is True and the inputs violate an assumption.
    """
    values = _checked_inputs(inputs, ("income_ok", "credit_ok", "guarantor", "bankrupt"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_eligible_value = _compute_output_eligible(values["income_ok"], values["credit_ok"], values["guarantor"], values["bankrupt"])
    return output_eligible_value


def evaluate(inputs: Mapping[str, bool], *, check_scope: bool = True) -> Dict[str, bool]:
    """Evaluate every output (declared order); each output is computed once.

    Required inputs: 'income_ok', 'credit_ok', 'guarantor', 'bankrupt'.
    Raises InputError for missing, undeclared, or non-bool inputs, and OutOfScopeError
    when check_scope is True and the inputs violate an assumption.
    """
    values = _checked_inputs(inputs, ("income_ok", "credit_ok", "guarantor", "bankrupt"), check_scope)
    if check_scope:
        _require_in_scope(values)
    output_eligible_value = _compute_output_eligible(values["income_ok"], values["credit_ok"], values["guarantor"], values["bankrupt"])
    return {
        "eligible": output_eligible_value,
    }

PostgreSQL

-- Tangibool PostgreSQL export (tangibool.postgresql/1)
-- Target: PostgreSQL 16 or later with typed boolean inputs; tested with PostgreSQL 18.3 (PGlite 0.5.8).
-- Not a claim of compatibility with other SQL dialects.
-- Model: not stated
-- Source revision: reference
-- Interpretation status: not stated
-- Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1)
-- Logic digest: 40aeadff132d113e9694381e8ed2530396d9cd296f720e1c4c08565e8f3b69f4
-- Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
-- Mode: decision
--
-- Read-only: this file contains only WITH / VALUES / SELECT statements. No DDL, DML,
-- procedures, credentials, network access, or database changes. The expressions are pure
-- Boolean expressions, so their result does not depend on SQL evaluation order.
--
-- NON-NULL INPUT PRECONDITION: NULL is not a truth value of this rule. Every fragment below
-- is defined only when every input column it reads is a non-NULL boolean. The guard template
-- (section 2) marks any NULL input as 'invalid_input' and any row that violates the
-- assumptions as 'out_of_scope', and fills output columns only for 'ok' rows; a NULL in an
-- output column of another row is an envelope marker, not a third truth value. Never wrap
-- the fragments in COALESCE(..., FALSE) or otherwise choose a meaning for unknown inputs.
-- This evaluates the supplied model; it does not show that the model is a complete, legal,
-- or safe statement of a real-world rule.
--
-- Input columns (stable ID: name -> column):
--   v_6cab253eb6ee: income_ok -> "in_income_ok"
--   v_d61ccf3c5e24: credit_ok -> "in_credit_ok"
--   v_a90721d191bc: guarantor -> "in_guarantor"
--   v_c9e9282e85b4: bankrupt -> "in_bankrupt"
-- Output columns:
--   f_a3631a5ff6db: eligible -> "out_eligible"
-- Assumptions (scope K; exported separately and never folded into an output):
--   none; every complete input row is in scope.

-- ----------------------------------------------------------------------------------------------
-- 1. Expression fragments (parenthesised; non-NULL boolean inputs required)
-- ----------------------------------------------------------------------------------------------
--
-- [out_eligible] output eligible (stable ID f_a3631a5ff6db)
--   eligible := income_ok AND (credit_ok OR guarantor) AND NOT bankrupt
--   PRECONDITION: "in_income_ok", "in_credit_ok", "in_guarantor", "in_bankrupt" are non-NULL booleans.
--   ("in_income_ok" AND ("in_credit_ok" OR "in_guarantor") AND (NOT "in_bankrupt"))
--

-- ----------------------------------------------------------------------------------------------
-- 2. Guard template (read-only SELECT over an inputs list)
-- ----------------------------------------------------------------------------------------------
-- Replace the example rows between the markers with your own rows, or replace the inputs
-- CTE with a SELECT that provides the boolean columns listed in the header. status is
-- 'invalid_input' when any input is NULL, 'out_of_scope' when the assumptions are false, and
-- 'ok' otherwise; output columns are filled only for 'ok' rows (NULL elsewhere is an
-- envelope marker). Assumption columns are shown for rows with complete inputs.
WITH inputs ("row_id", "in_income_ok", "in_credit_ok", "in_guarantor", "in_bankrupt") AS (
  -- tangibool:example-rows:begin
  VALUES
    (1, FALSE, FALSE, FALSE, FALSE),
    (2, FALSE, FALSE, FALSE, TRUE),
    (3, FALSE, FALSE, TRUE, FALSE),
    (4, FALSE, FALSE, TRUE, TRUE),
    (5, FALSE, TRUE, FALSE, FALSE),
    (6, FALSE, TRUE, FALSE, TRUE),
    (7, FALSE, TRUE, TRUE, FALSE),
    (8, FALSE, TRUE, TRUE, TRUE),
    (9, TRUE, FALSE, FALSE, FALSE),
    (10, TRUE, FALSE, FALSE, TRUE),
    (11, TRUE, FALSE, TRUE, FALSE),
    (12, TRUE, FALSE, TRUE, TRUE),
    (13, TRUE, TRUE, FALSE, FALSE),
    (14, TRUE, TRUE, FALSE, TRUE),
    (15, TRUE, TRUE, TRUE, FALSE),
    (16, TRUE, TRUE, TRUE, TRUE),
    (17, NULL::boolean, FALSE, FALSE, FALSE)
  -- tangibool:example-rows:end
),
checked AS (
  SELECT
    inputs.*,
    CASE
      WHEN "in_income_ok" IS NULL OR "in_credit_ok" IS NULL OR "in_guarantor" IS NULL OR "in_bankrupt" IS NULL THEN 'invalid_input'
      ELSE 'ok'
    END AS "status"
  FROM inputs
)
SELECT
  "row_id",
  "in_income_ok",
  "in_credit_ok",
  "in_guarantor",
  "in_bankrupt",
  "status",
  CASE WHEN "status" = 'ok' THEN ("in_income_ok" AND ("in_credit_ok" OR "in_guarantor") AND (NOT "in_bankrupt")) END AS "out_eligible"
FROM checked
ORDER BY "row_id";

SMT-LIB

; Tangibool SMT-LIB export (tangibool.smtlib/1)
; Profile: SMT-LIB 2.6 Boolean subset (QF_UF): Bool constants, define-fun, let, push/pop, assert, check-sat, echo.
; Tested with Z3 5.1.0 (z3-solver 5.2.0, WebAssembly build).
; Model: not stated
; Source revision: reference
; Interpretation status: not stated
; Exporter: 1.0.0 (engine 1.0.0; semantics tangibool-semantics/1)
; Logic digest: 40aeadff132d113e9694381e8ed2530396d9cd296f720e1c4c08565e8f3b69f4
; Assumptions digest: d58ee1ad38e23e5093e4baa1047b5ae70d6972e08f555e34ac3fa16df63c4a78
; Mode: decision
;
; Reading results: each query block states what sat and unsat mean for its question. A sat
; answer comes with a witness or counterexample (ask for it with (get-model) after sat only);
; unsat settles that scoped query and is solver-checked, not an independently checked proof
; certificate; unknown settles nothing. Blocks are scoped with push/pop, so assertions never
; accumulate across questions. This checks the supplied model, not whether it is a complete,
; legal, or safe statement of a real-world rule.
;
; Symbols (stable ID: name -> symbol):
;   v_6cab253eb6ee: income_ok -> in_income_ok
;   v_d61ccf3c5e24: credit_ok -> in_credit_ok
;   v_a90721d191bc: guarantor -> in_guarantor
;   v_c9e9282e85b4: bankrupt -> in_bankrupt
;   f_a3631a5ff6db: eligible -> out_eligible

(set-option :produce-models true)
(set-info :smt-lib-version 2.6)
(set-logic QF_UF)

; ---- Inputs (declared order)
(declare-const in_income_ok Bool) ; income_ok
(declare-const in_credit_ok Bool) ; credit_ok
(declare-const in_guarantor Bool) ; guarantor
(declare-const in_bankrupt Bool) ; bankrupt

; ---- Outputs (dependencies first)
; output eligible (stable ID f_a3631a5ff6db): income_ok AND (credit_ok OR guarantor) AND NOT bankrupt
(define-fun out_eligible () Bool (and in_income_ok (or in_credit_ok in_guarantor) (not in_bankrupt)))

; ---- Assumptions (the scope K; never folded into an output)
; K: conjunction of the enabled, confirmed assumptions (true when there are none)
(define-fun K () Bool true)

; ---- Definitions only: nothing is asserted and no check is run.
; Add (assert ...) and (check-sat) inside (push 1) ... (pop 1) to ask a question.

LaTeX

eligible = \mathit{income\_ok} \cdot \left(\mathit{credit\_ok} + \mathit{guarantor}\right) \cdot \overline{\mathit{bankrupt}}

The common slip

Keeping only the rows where the rule is TRUE. A suite of eligible applicants proves that the rule admits; it never checks that it refuses, and the bankrupt pair, eligible to ineligible by one condition, is the row that matters most.

Questions

Why not test every row?

For four conditions you can, and the export makes it cheap. The method is for the rules where you cannot, and the pairs are the rows you would regret dropping: each one is the only evidence that some condition does anything.

How do expected results get into the tests?

From the truth table, which is the rule made exhaustive, and the rule from its requirement. The Python export evaluates the model; it does not decide what the model should say.

Testing and rules

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.